Both Ryan Callahan for New York and Jakob Silfverberg for Ottawa scored in the shootout's second round, after which a goaltending duel kept the penalty shots coming.
The goal by Daugavins gave the Senators their third straight victory and made a winner out of Ben Bishop.
Bishop came into the game after starting Ottawa goaltender Craig Anderson suffered an apparent leg injury in a collision with New York's Chris Kreider 1:43 into the third period. The backup goaltender gave up two goals in the third, but then stopped six out of seven in the shootout.
New York's two third-period goals came 49 seconds apart to put the Rangers in front, but Mika Ziebanejad scored on the power play for the Senators to force overtime.
Henrik Lundqvist made 35 saves for the Rangers, who have lost two in a row after winning four out of five.